Necklace ring

In mathematics, the necklace ring is a ring introduced by Template:Harvs to elucidate the multiplicative properties of necklace polynomials.

Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences $(a_1,a_2,...)$ of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of $(a_1,a_2,...)$ and $(b_1,b_2,...)$ has components

$c_n=\sum _{[i,j]=n}(i,j)a_i{b}_j$

where $[i,j]$ is the least common multiple of $i$ and $j$, and $(i,j)$ is their greatest common divisor.

This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence $(a_1,a_2,...)$ with the power series $\prod _{n\ge 0}(1-t^n{)}^{-a_n}$.

See also

• Witt vector

References

• Template:Cite book
• Template:Cite journal